On the Weyl anomaly for chiral fermions

TL;DR

The paper investigates whether a parity-odd component of the Weyl (conformal) anomaly for a chiral fermion in a gravitational background exists, prompted by claims of a Pontryagin-density term. It employs a manifestly real Lagrangian and Pauli-Villars regularization to regulate UV divergences without the ambiguities of dimensional regularization for $\gamma_5$. The main result is that all parity-odd contributions cancel at one loop, so the Weyl anomaly is parity-even and built from standard invariants like the Euler density $E_4$ and Weyl-squared $W^2$, with no Pontryagin-term. This resolves concerns about unitarity and clarifies the one-loop structure of the conformal anomaly for chiral fermions in a gravitational background; no simple topological interpretation via an index theorem is established.

Abstract

We compute the parity-odd part of the Weyl anomaly for chiral fermions in a background gravitational field. We start from a manifestly real form of the Lagrangian (that is, not only real up to a total derivative), and we regularize it by means of Pauli-Villars fermions. All parity-odd terms in the anomaly cancel in the integrand, so that the result of the anomaly is necessarily parity-even.